A bare interval object may be nothing more than such a diagram. If CC admits sufficiently many limits and colimits, then from this alone a lot of structure derives. The precise definition of further structure and property imposed on an interval object varies with the intended context and applications.

Notably in a large class of applications the interval object in CC supposed to be the right structure to ensure

For instance the choice C=C =Top and I=[0,1]I = [0,1] should be an instance of a category with interval object, and the fundamental algebraicn-groupoidΠn(X)\Pi_n(X) obtained for any topological space XX from this data should be the fundamental nn-groupoid as a Trimble n-category.

We give two very similar definitions that differ only in some extra assumptions.

The first one was used by Berger and Moerdijk to generalize the Boardman–Vogt resolution of topological operads to more general operads.

In homotopical categories

Trimble interval object

The following definition is strongly related to the notion of Trimble omega-category where the interval object gives the internal hom [I,X][I,X] the structure of an operad giving (by induction) the model of an A∞A_\infty-category structure on

For V=C=TopV = C = Top with its standard model structure the standard topological closed interval I:=[0,1]I := [0,1] with pt→σ,τIpt \stackrel{\sigma, \tau}{\to}I the maps to 0 and 1, respectively. This is the case described in detail at Trimble n-category.

For V=ωCatV = \omega Cat the category of strict omega-categories the first oriental, the 1-globeI={a→b}I = \{a \to b\} is an interval object. In this strict case in fact all hom objects are already equal to the point pt[I,I∨n]pt=pt{}_{pt}[I, I^{\vee n}]_{pt} = pt and

is a strict co-category internal to ω\omegaCat. In this case, for XX any ω\omega-category the A∞A_\infty-category Π1(X)\Pi_1(X) is just an ordinary category, namely the 1-category obtained from truncation of XX. Similarly, probably Πω(X)=X\Pi_\omega(X) = X in this case.

But another possible choice is to let I=ℝI = \mathbb{R} be the whole real line, but still equipped with the two maps 0,1:*→ℝ0,1 : {*} \to \mathbb{R}, that hit the 0∈ℝ0 \in \mathbb{R} and 1∈ℝ1 \in \mathbb{R}, respectively.

Either of these two examples will do in the following discussion. The second choice is to be thought of as obtained from the first choice by adding “infinitely wide collars” at both boundaries of [0,1][0,1]. While *→0[0,1]←1*{*} \stackrel{0}{\to}[0,1]
\stackrel{1}{\leftarrow} {*} may seem like a more natural choice for a representative of the idea of the “standard interval”, the choice *→0ℝ←1*{*} \stackrel{0}{\to} \mathbb{R}
\stackrel{1}{\leftarrow} {*} is actually more useful for many abstract nonsense constructions.

But since it is hard to draw the full real line, in the following we depict the situation for the choice I=[0,1]I = [0,1].

The two face maps δ1*→I\delta_1 {*} \to I and δ0:*→I\delta_0 : {*} \to I pick the boundary points in the obvious way. The unique degeneracy map σ0:I→*\sigma_0 : I \to {*} maps all points of the interval to the single point of the point.

But the three face maps δi:I→I×I\delta_i : I \to I\times I of the cosimplicial object ΔI\Delta_I constructed above don’t regard the full square here, but just a triangle sitting inside it, in that pictorially they identify (ΔI1=I)(\Delta_I^1 = I)-shaped boundaries in I×II \times I as follows:

𝔸1\mathbb{A}^1-homotopy theory

Fundamental ∞\infty-categories induced from intervals

The interest in interval objects is that various further structures of interest may be built up from them. In particular, since picking an interval object II is like picking a notion of path, in a category with interval object there is, under mild assumptions, for each object XX an infinity-categoryΠI(X)\Pi_I(X) – the fundamental ∞\infty-category of XX with respect to II – whose k-morphisms are kk-fold II-paths in XX.